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\begin{document}

$$
F(x) = \frac12 \|Ax-b\|_2^2 + \lambda \|x\|_1
$$

 

$$
F(x) = \frac12 \|Ax-b\|_2^2 + \frac\lambda2 \|x\|_2^2
$$

$$
F(x) = \frac12 \sum_{j=1}^m  \max\{0, 1-y_j A_{j}^T x   \}^2 + \lambda \|x\|_1
$$

$$
F(x) = \frac12 \sum_{j=1}^m  \max\{0, 1-y_j A_{j}^T x   \}^2 + \frac\lambda2 \|x\|_2^2
$$

$$
F(x) =  \sum_{j=1}^m  \log (1+e^{-y_j A_{j}^T x }) + \lambda \|x\|_1
$$

$$
F(x) =  \sum_{j=1}^m  \log (1+e^{-y_j A_{j}^T x }) + \frac\lambda2 \|x\|_2^2
$$

$$
F(x) = \frac1m \sum_{j=1}^m  \max\{0, 1-y_j A_{j}^T x   \}+ \frac\lambda2 \|x\|_2^2
$$

\begin{itemize}
 \item $A \in {R}^{m \times n}$
 \item $A_{j} \in R^n$ is a column vector consisting of $j$-th row of matrix $A$
\end{itemize}




\end{document}
